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Assume I have a system of linear equations $Au = b$ where A is some $M\times N$ matrix, u is $N\times 1$ unknown variables vector, and b is also $N\times 1$$M\times 1$. Now, further assume that I know that some elements of $u$ are linearly dependent on the other elements of $u$. I.e., we can write $u$ as $u=[u_1, u_2]$ where $u_1$ is $N_1\times 1$, and $u_2$ is $N_2 \times 1$ and $N_1+N_2=N$ and it is known that $u_1 = Cu_2$, where $C$ is some unknown matrix. Is there any way that this knowledge (that we know that some elements of u are dependent and we know which elements, but we don't know the dependency matrix C) can help me improve the complexity of solving the system of equations $Au=b$? I.e., is there any way it can simplify the complexity of the gaussian elimination (or any other algorithm for solving a system of linear equations)?

Thanks!

Assume I have a system of linear equations $Au = b$ where A is some $M\times N$ matrix, u is $N\times 1$ unknown variables vector, and b is also $N\times 1$. Now, further assume that I know that some elements of $u$ are linearly dependent on the other elements of $u$. I.e., we can write $u$ as $u=[u_1, u_2]$ where $u_1$ is $N_1\times 1$, and $u_2$ is $N_2 \times 1$ and $N_1+N_2=N$ and it is known that $u_1 = Cu_2$, where $C$ is some unknown matrix. Is there any way that this knowledge (that we know that some elements of u are dependent and we know which elements, but we don't know the dependency matrix C) can help me improve the complexity of solving the system of equations $Au=b$? I.e., is there any way it can simplify the complexity of the gaussian elimination (or any other algorithm for solving a system of linear equations)?

Thanks!

Assume I have a system of linear equations $Au = b$ where A is some $M\times N$ matrix, u is $N\times 1$ unknown variables vector, and b is $M\times 1$. Now, further assume that I know that some elements of $u$ are linearly dependent on the other elements of $u$. I.e., we can write $u$ as $u=[u_1, u_2]$ where $u_1$ is $N_1\times 1$, and $u_2$ is $N_2 \times 1$ and $N_1+N_2=N$ and it is known that $u_1 = Cu_2$, where $C$ is some unknown matrix. Is there any way that this knowledge (that we know that some elements of u are dependent and we know which elements, but we don't know the dependency matrix C) can help me improve the complexity of solving the system of equations $Au=b$? I.e., is there any way it can simplify the complexity of the gaussian elimination (or any other algorithm for solving a system of linear equations)?

Thanks!

Source Link
MRm
MRm
  • 183
  • 4

Linear system of equations with prior knowledge on linear dependency of some unknowns

Assume I have a system of linear equations $Au = b$ where A is some $M\times N$ matrix, u is $N\times 1$ unknown variables vector, and b is also $N\times 1$. Now, further assume that I know that some elements of $u$ are linearly dependent on the other elements of $u$. I.e., we can write $u$ as $u=[u_1, u_2]$ where $u_1$ is $N_1\times 1$, and $u_2$ is $N_2 \times 1$ and $N_1+N_2=N$ and it is known that $u_1 = Cu_2$, where $C$ is some unknown matrix. Is there any way that this knowledge (that we know that some elements of u are dependent and we know which elements, but we don't know the dependency matrix C) can help me improve the complexity of solving the system of equations $Au=b$? I.e., is there any way it can simplify the complexity of the gaussian elimination (or any other algorithm for solving a system of linear equations)?

Thanks!